All functions are (locally) $s$-harmonic (up to a small error) - and applications

Abstract: The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences.